The following construction of \(\mathrm {Spin}(7)\) , and its eight-dimensional spin representation, are developed in [1]}. Consider the octonion algebra \(\mathbb {O}\)  and the 4-form \(\phi \in \bigwedge ^4 \mathbb {O}^*\)  defined by \(\phi (a,b,c,d) = \langle a,b\times c \times d\rangle \) , where the triple cross product is by definition \(b\times c \times d = \frac{1}{2} \bigl ( b(\bar{c} d) - d(\bar{c} b) \bigr )\)  and \(\bar{c}\)  is the octonionic conjugate of \(c\) . Then \(\mathrm {Spin}(7)\)  is precisely the subgroup of \(\mathrm {SO}(8)\)  fixing \(\phi \) . In terms of coordinates \(x_1,x_2,\dots ,x_8\)  on \(\mathbb {R}^{0|8}\) , \(\phi \)  corresponds to the function
\(\phi = x_{1234} + x_{1256} + x_{1278} + x_{1357} - x_{1368} - x_{1458} - x_{1467} \\\hphantom{\phi =}{} + x_{5678} + x_{3478} + x_{3456} + x_{2468} - x_{2457} - x_{2367} - x_{2358} ,\) 
